This algebra 2 inference decisions lesson explains statistical experiments, simulation and decisions in one printable page, with definitions, a method and a worked example.
Statistical decisions compare results with what chance alone would produce. A simulation repeats a random process many times to test whether a result is consistent with a model. In a randomized experiment, a difference is significant when re-randomizing rarely matches it. Expected value and fair random choices apply probability to decisions.
Algebra 2 Inference Decisions Lesson: key facts
- Standards S-IC.A.2, S-IC.B.5, S-IC.B.6, S-MD.B.6 and S-MD.B.7: simulation, experiments, reports and decisions.
- A result is inconsistent with a model when simulated sets rarely produce one that extreme (often under 5%).
- Random assignment lets a significant difference be credited to the treatment, not to luck.
- A choice is fair when every option has the same probability; expected value is the sum of payoff times probability.
- Check a report’s sample, assignment, size and presentation before trusting its claim.
How do you decide whether a result fits a model?
A model is a probability claim, such as “this die is fair.” A result is consistent with the model if it could easily happen by chance when the model is true. To check, run a simulation: repeat the random process many times with random digits or software and record each outcome.
- State the model and the expected result.
- Simulate many sets with the same number of trials.
- Count the sets as extreme as the real result.
- If that share is below about \(5\%\), the model is doubtful.
Liam rolled a die \(60\) times and got \(20\) sixes. A fair die gives about \(60\cdot\frac{1}{6}=10\). In \(200\) simulated sets, only 1 set had \(20\) or more sixes, so the die is probably not fair.
How do you compare two treatments?
In a randomized experiment, chance assigns subjects to treatments. Maya grew \(12\) seedlings, \(6\) with fertilizer A and \(6\) with B. The means were \(16.5\text{ cm}\) and \(13\text{ cm}\), a difference of \(3.5\text{ cm}\).
To see whether luck alone could do this, pool the \(12\) heights and split them into two groups of \(6\) in every possible way. Only \(5\) of the \(924\) splits gave a difference of \(3.5\text{ cm}\) or more, about \(0.5\%\). A difference this rare is statistically significant.
How do you evaluate a report based on data?
A random sample supports claims about a population. Random assignment supports cause and effect. A study that only observes groups shows association. Small samples, loaded wording and graphs with a cut-off axis can all mislead.
How can probability make fair and wise decisions?
A choice is fair when every option has the same probability. To pick one of three friends, roll a die: \(1\) or \(2\) for the first, \(3\) or \(4\) for the second, \(5\) or \(6\) for the third. The expected value is the sum of each payoff times its probability.
Noah pays \(\$3\) to play a game that wins \(\$20\) with probability \(0.1\). His expected net result is \(0.1\cdot 17+0.9\cdot(-3)=-\$1.00\) per play, so the game favors the school, not Noah.
Statistical Experiments, Simulation and Decisions: frequently asked questions
What does it mean for a result to be consistent with a model?
It means the result could easily occur by chance if the model were true. If simulations of the model often produce results this extreme, the model is not contradicted.
What does statistically significant mean?
A difference is statistically significant when it would rarely happen by chance alone, often less than 5% of the time. It does not mean the difference is large or important.
Why is random assignment so important in an experiment?
Chance spreads other differences among the groups about evenly. A large difference in outcomes is then unlikely to be luck or a hidden factor, so the treatment is the likely cause.
How do you make a fair decision with a die or coin?
Assign the options equally likely outcomes, such as two die faces for each of three people. If some outcomes are left over, roll again.
What is expected value and how does it help decide?
Expected value is the average result over many repeats: multiply each payoff by its probability and add. A positive expected net result favors playing, and a negative one does not.
How to use this algebra 2 inference decisions lesson
Use this algebra 2 inference decisions lesson before the worksheet, then check what you learned with the quiz.
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Algebra 2 Statistical Experiments, Simulation and Decisions lesson
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